RHS-to-RHS Axially Loaded X -Connections near an … Axially Loaded X -Connections near an Open...

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RHS-to-RHS Axially Loaded X-Connections near an Open

Chord End

Journal: Canadian Journal of Civil Engineering

Manuscript ID cjce-2017-0148.R1

Manuscript Type: Article

Date Submitted by the Author: 06-Jun-2017

Complete List of Authors: Fan, YuJing; University of Toronto, Civil Engineering Packer, Jeffrey; University of Toronto, Civil Engineering

Is the invited manuscript for consideration in a Special

Issue? : N/A

Keyword:

experimental research < type of paper to review, structure - steel <

Struct.Eng. & Constr.Mate, code formul & safe cons < Struct.Eng. & Constr.Mate

https://mc06.manuscriptcentral.com/cjce-pubs

Canadian Journal of Civil Engineering

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RHS-to-RHS Axially Loaded X-Connections near an Open Chord End 1

by 2

YuJing Fan and Jeffrey A. Packer* 3

Department of Civil Engineering, University of Toronto, 35 St. George Street, Toronto, ON, M5S 1A4, Canada 4

5

Abstract 6

In all truss-type welded hollow structural section connection design procedures, the chord member is 7

assumed continuous on both sides of the branch. New limits of applicability have recently been advocated, 8

which specify minimum end distances from an open chord end to achieve the full connection strength. To 9

investigate their suitability for rectangular hollow section (RHS) connections, an experimental program 10

consisting of 12 RHS-to-RHS X-connections subject to branch axial compression was undertaken. For 11

connections near a chord end, a modified yield line mechanism controlled. A proposed analytical model is 12

derived, and by evaluating against the experimental results, is shown to accurately predict the yield loads. An EN 13

1993-1-8 amendment, transcribed from circular hollow section connections, is deemed excessively conservative, 14

and a closed-form solution for the end distance, presented herein, is recommended for use with RHS 15

connections. Alternatively, providing a cap plate is determined to be an effective stiffening method. 16

17

Keywords 18

Steel structures, hollow structural sections, rectangular hollow sections, truss connections, welded joints, chord 19

end effects, static loading, analytical modelling 20

* Corresponding author, email: [email protected] 21

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Introduction 1.22

Specific research on truss-type, welded hollow structural section (HSS) connections has been conducted 23

since the 1960s, resulting in connection design recommendations published in North America (Packer and 24

Henderson 1997; AISC 2010) and internationally (CEN 2005; Packer et al. 2009; ISO 2013). All of these HSS 25

design recommendations presume that the chord member is continuous for a considerable distance beyond the 26

connection region. There is no established guidance for the case when an HSS branch member is near an open 27

end of an HSS chord member in a girder or truss, which is a problem encountered with trusses supported beneath 28

the bottom chord (such as shown in Fig. 1) or with free chord ends unconnected to another member (such as 29

Warren truss bottom chords where the truss is suspended from connections to the ends of the top chord). In this 30

situation, designers typically resort to strengthening the chord ends, such as capping the chord with an end plate 31

or reinforcing the chord face or walls with stiffening plates. However, there should be a minimum end distance, 32

emin, from an open chord end at which full connection strength can be achieved for an unreinforced connection. 33

34 Fig. 1. RHS-to-RHS connection near a truss open end, with chord stiffened 35

Limited research on HSS chord end effects has been performed for circular hollow section (CHS) welded 36

connections. The numerical investigations on CHS-to-CHS T- and X-connections by van der Vegte and Makino 37

(2006; 2010) concluded that chord boundary conditions were not influential if the chord length was greater than 38

10 times the CHS diameter (10d0) for all connection geometries studied. It was shown that the required chord 39

length was primarily a function of the chord wall slenderness (2γ), where less stiffness was provided to limit 40

chord plastification if the connection had thin chord walls compared to thick chord walls. For shorter chord 41

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lengths, significant differences in the strengths of connections with fixed (capped with end plates) versus free 42

end conditions were determined. Recent numerical studies on transverse branch plate-to-CHS T- and X-43

connections by Voth and Packer (2012a; 2012b) also substantiated these results. 44

The recommendations from this research may be employed for the design of isolated connections in 45

experimental or numerical tests, where the behaviour may be different from that of connections which are part of 46

a framed structure. In addition, they are the basis for the minimum end distance proposed for EN 1993-1-8, in 47

the following amendments to Clause 7.1.2 (9): 48

“For joints with a chord end not connected to other members, the chord end shall be at a distance of at least 49

(2γ/10)d0 from the heel or toe of the closest brace, with a minimum of 2.5d0. For RHS chords, substitute d0 by the 50

largest of b0 or h0. Otherwise, the end shall be welded to a cap plate with a thickness of at least 1.5t0, at a 51

minimum distance of 0.5d0(1-β) or 0.5b0(1-β) from the brace toe or heel of the joint.” 52

This end distance requirement is illustrated in Fig. 2. 53

54 Fig. 2. Proposed end distances in EN 1993-1-8 (shown for CHS) 55

The (2γ/10)d0 rule which is derived empirically for CHS chords, with a minimum value of 2.5d0 included for 56

good engineering practice, is transcribed for rectangular hollow section (RHS) chords by simply replacing the 57

outside dimensions. Specifically, substituting the CHS diameter (d0) by the largest of the RHS width and depth 58

(b0 and h0) and accounting for the applicable 2γ values lead to a large required end distance of 2.5b0 or 2.5h0 (the 59

minimum stipulated, as shown in Fig. 2) up to 4b0 or 4h0 (for the maximum 2γ of 40, as permitted by ISO (2013) 60

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for RHS). However, this end-effects rule accommodates the propagation of ovalization of the complete chord 61

member that occurs in CHS connections, whereas with RHS connections chord member distortion is generally 62

more localized and largely confined to the chord connecting face. Also, the research on chord end effects, on 63

which the EN 1993-1-8 formula is based, considered connections that were symmetrical about the branch, while 64

the behaviour of connections with the branch near a chord end may be different. These assumptions indicate that 65

the estimated end distances may be very conservative, thus further research is needed to develop end distance 66

requirements specific to RHS chord members. 67

If the branch needs to be closer to a chord end, the use of a cap plate is specified. For this case, the 68

respective minimum distance of 0.5d0(1-β) and 0.5b0(1-β) for CHS and RHS sections is equal to the distance 69

between the branch and the chord side walls, to ensure a similar stiffness around the branch. While this end 70

distance may be necessary for fabrication details, it should not be an issue for strength. The minimum cap plate 71

thickness of 1.5t0 is based on engineering judgement so that it has sufficient stiffness relative to the chord 72

sidewalls (i.e. the ratio of the cap plate to chord sidewall stiffness is (1.5t0)3/t0

3=3.38) to restrain local chord 73

deformation and develop the required connection strength. 74

Analytical Study (Yield Line Analyses) 2.75

Analytical yield line analysis has been used to determine the capacity of flexible HSS connections. When a 76

connection is remote from the RHS chord ends (defined as a “regular” connection), the analytical solution for the 77

yield load PY (Eq. (1)), associated with the chord face yielding limit state, can be derived based on a symmetric 78

flexural yield line mechanism developing in the chord connecting face (Fig. 3). Qf is a function to account for the 79

effect of chord normal stress, which is less than or equal to one. Eq. (1) has been widely adopted in design 80

recommendations for RHS-to-RHS T-, Y-, and X-connections. 81

(1) ( )

20 0 1

1

0 1

2sin 4 1

1 sin

y

Y f

f t hP Q

b

θ = + −β −β θ

82

In Eq. (2), the full dimensions h1, b1, b0 and β in Eq. (1) are replaced by the more accurate effective 83

dimensions h1/, b1

/, b0

/ and β

/ where: 84

• h1 is increased to account for the fillet weld size, or h1/ = h1/sinθ1 + 2w 85

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• b1 is increased to account for the fillet weld size, or b1/ = b1 + 2w 86

• b0 is decreased to reflect plastic hinges forming in the centre of the chord walls, or b0/ = b0 – t0 87

• branch-to-chord width ratio β = b1/b0 is increased to β/ = b1

//b0

/ 88

(2) ( )

2 /0 0/ /1

1 //0

2sin 4 1

1

y

Y f

f t hP Q

b

θ = + −β

−β 89

90 Fig. 3. Yield line mechanism for a regular RHS-to-RHS axially-loaded X-connection 91

If β approaches unity, the yield load tends to infinity and this limit state is not likely to be critical. β ≤ 0.85 92

thus represents a practical upper limit for the application of the yield line solution. 93

When a connection is close to a RHS chord end (termed an “offset” connection), there may not be adequate 94

lengths to develop the traditional yield line mechanism. Potential asymmetric flexural yield line mechanisms are 95

therefore proposed and investigated for their ability to predict the yield loads of the offset connections, where 96

solutions for emin can also be derived by comparison with Eqs. (1) or (2). 97

A first possible flexural yield line mechanism no. 1 (Fig. 4) is studied. The analytical yield load solution, for 98

a known end distance e from the branch edge to an open chord end, is given by Eq. (3). 99

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(3) ( )

20 0 1

,1 1

0 0 1

22sin 2 1

1 sin

y

Y f

f t heP Q

b b

θ = + + −β −β θ

100

101 Fig. 4. Yield line mechanism No. 1 for an offset RHS-to-RHS axially-loaded X-connection 102

It can be shown that this postulated yield line mechanism controls and predicts a lower yield load compared 103

to the traditional yield line mechanism (Eq. (1)) if the end distance is less than: 104

(4) min 0 1e b= −β 105

If e ≥ emin, end distance will not be critical and the yield load for the regular T-, Y-, or X-connection 106

represents the minimum connection strength. 107

In Eq. (5), the full dimensions h1, b1, b0, β and e in Eq. (3) are substituted with effective dimensions h1/, b1

/, 108

b0/, β

/ and e

/, where e is reduced to account for the fillet weld size, or e

/ = e – w. 109

(5) ( )

2 //0 0/ /1

,1 1 / //0 0

22sin 2 1

1

y

Y f

f t heP Q

b b

θ = + + −β

−β 110

The associated minimum end distance is then: 111

(6) / /min 0 1e b w= −β + 112

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A second possible flexural yield line mechanism no. 2 (Fig. 5) is also investigated in the yield line 113

predictions (Eq. (7), where m is the slope of the inclined yield line from the branch side to the open chord end). 114

This mechanism applies only if e ≤ mb1/2 or e ≤ mb1//2 + w (i.e. if the yield lines have not converged). 115

(7) ( )

20 0 1

,2 1 20 0 1

21 1sin 2 2 1

1 sin 2

y

Y f

f t heP Q

b b mm

θ = + + + −β + −β θ 116


This approximate yield line mechanism governs over the regular yield line mechanism (Eq. (1)), if the end 119

distance is less than: 120

(8) ( ) ( )2min 0 1 1 4 1 1 2e b m m = −β − + 121

If e ≥ emin, end distance will not be critical. 122

In Eq. (9), the full dimensions h1, b1, b0, β and e in Eq. (7) are substituted with effective dimensions h1/, b1

/, 123

b0/, β

/ and e

/: 124

(9) ( )

2 //0 0/ /1

,2 1 2 / //0 0

21 1sin 2 2 1

21

y

Y f

f t heP Q

mm b b

θ = + + + −β + −β

125

The corresponding minimum end distance is then: 126

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(10) ( ) ( )/ / 2min 0 1 1 4 1 1 2e b m m w = −β − + +

127

A third possible flexural yield line mechanism no. 3 (Fig. 6) is considered as well (Eq. (11), where x is the 128

distance required for the inclined yield lines from the branch side to the open chord end to converge). This 129

mechanism applies only if e > x or e > x/ + w (i.e. if the yield lines have converged). 130

(11) ( )

( )20 0 11

,3 1

0 0 1

122sin 2 1

1 sin 4

y

Y f

f t bheP Q

b b x

+ βθ = + + −β +

−β θ 131


Since mechanism no. 3 applies only for e > x or e > x/ + w, the yield load predicted by this yield line 134

mechanism in the applicable range will generally be greater than the yield load predicted by the regular yield line 135

mechanism. Thus, it is not a critical mechanism for end effects, and a minimum end distance does not apply. 136

In Eq. (12), the full dimensions h1, b1, b0, β, e and x in Eq. (11) are substituted with effective dimensions h1/, 137

b1/, b0

/, β

/, e

/ and x

/, where x is reduced to account for the fillet weld size, or x

/ = x – w. 138

(12) ( )

( )/ /2 //10 0/ /1

,3 1 / / //0 0

122sin 2 1

41

y

Y f

bf t heP Q

b b x

+ β θ = + + −β + −β

139

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The ratios of emin/b0, calculated for mechanisms nos. 1 and 2 using Eqs. (4) and (8), are plotted against β (i.e. 140

represented by the two lines) in Fig. 7, which shows that emin is always less than b0 for all values of β. Thus, 141

according to the yield line solution, a conservative end distance equal to the chord member width can be used for 142

a branch connected close to an open chord end. 143

However, as the minimum end distances are derived analytically, they have not been substantiated by 144

experimental evidence. The formulae are also only a function of β, the branch-to-chord width ratio, and do not 145

account for the possible influence of other connection parameters (e.g. 2γ, the chord slenderness ratio). Further, a 146

yield line mechanism, which is based on a chord face flexure limit state, may not always govern. For β > 0.85, 147

which is outside the yield line theory applicability, a sidewall failure limit state may control. 148

2.1. Ultimate Deformation Limit Applied to Tests 149

The connection capacity at the ultimate limit state can be determined from experimental or numerical load-150

displacement results as the lower of, as implemented by IIW (2012) and shown in Fig. 8: 151

(a) The maximum load resulting from ultimate failure if this load occurs at a lower displacement than (b); or 152

(b) The load corresponding to an ultimate deformation limit 153

A connecting face deformation equal to 3% of the RHS chord width (0.03b0) or 3% of the CHS chord 154

diameter (0.03d0) has generally been used as the ultimate deformation limit in (b), as proposed by Lu et al. 155

(1994). The applicability of this ultimate deformation limit for various RHS welded connections has previously 156

Fig. 7. emin/b0 vs. β based on postulated yield line mechanisms

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been investigated (Zhao 2000; Kosteski and Packer 2003a, 2003b; Kosteski et al. 2003). As the 3% b0 ultimate 157

deformation limit load N1,3% was shown to agree well with the analytical yield load PY for these regular 158

connections, it will also be used as a limit to the connection capacity and to validate the proposed yield line 159

formulae for the offset connections. 160

161

Fig. 8. Failure criteria (adapted from Wardenier et al., 2010) 162

Experimental Program 3.163

An experimental program was undertaken to investigate the behaviour of connections near an open chord 164

end, evaluate the proposed yield line models, and assess the EN 1993-1-8 amendments for RHS welded 165

connections. 166

3.1. Test Specimens and Measured Dimensions 167

Twelve isolated, square HSS-to-HSS X-connection tests were performed, with the branch loaded in axial 168

compression and centred laterally on the chord. The connections were designed with “thick branches welded to 169

thin chords” in order to generate connection failure before reaching the branch member capacity. This is against 170

tubular design principles in practice, when the objective would be to avoid a connection failure. All chord and 171

branch members were made of cold-formed HSS sections. The branch members were produced to ASTM A847 172

(2014), which is a weathering steel grade, but has similar strength properties as ASTM A500 Grade C. The cap 173

plate materials were made to CSA G40.21 (2013) Grade 350W. Two chord sizes (HSS 203x203x6.35 and HSS 174

203x203x9.53) and one branch size (HSS 102x102x9.53) were selected, enabling one branch-to-chord width 175

ratio (nominal β = 0.5) and two chord slenderness ratios (nominal 2γ = 32 and 21). An empirical parametric 176

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study was performed with the clear end distance (e) from the HSS branch to the open chord end being the 177

primary variable (Fig. 9 (a)), and ranging from almost zero (but still sufficient to accommodate the welds) up to 178

slightly over min 0 1 144mme b= −β = (the yield line solution for mechanism no. 1, Eq. (4)). The chord length 179

on the other side of the branch was made sufficiently long to minimize the influence of chord end conditions and 180

was estimated from the EN 1993-1-8 formula. The effect of welding a chord end cap plate, for when the branch 181

is very close to the chord end (Fig. 9 (b)), was also investigated. The end distance and plate thickness for this 182

case satisfy the EN 1993-1-8 requirements. Since the top of the branch was uniformly compressed, shear lag was 183

not a problem and a branch height of four times the branch width (4b1) was considered adequate. Details of the 184

test specimens, including the measured dimensions, are tabulated in Table 1. 185

A 0.045’’ (1.2mm) diameter AWS A5.18 E70C-6M H4 (CSA W48 E491C-6M-H4) gas-shielded metal-186

cored wire (Metal-Cor 6-L) with a minimum specified tensile strength of 70ksi (480MPa) was used to weld the 187

test specimens. The branches, with both ends machined normal to the member axis, were welded to the flat faces 188

of the chord members with a continuous fillet weld around the branch perimeter. Two specimens had a 1-in. 189

thick cap plate welded to the chord end closest to the branch, with the weld ground flush at the bottom. Single-190

pass welds were used and the average horizontal leg size for each series of six specimens using the 2γ=34 and 23 191

HSS chords was determined to be 7.8mm and 7.9mm, respectively. 192

(a) without a cap plate (b) with a cap plate

Fig. 9. Square HSS-to-HSS X-connection test specimens 193

194

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Table 1. Test specimens and measured dimensions 195

Specimen Specimen 2γ

End distance Chord length

no. identification e [mm] l0 [mm]

1 X-0.5-32-25O 34 24 827

2 X-0.5-32-55O 34 55 764

3 X-0.5-32-100O 34 100 796

4 X-0.5-32-160O 34 160 875

5 (control) X-0.5-32-700O 34 714 1530

6 (+cap plate) X-0.5-32-55C 34 55 759

7 X-0.5-21-25O 23 25 687

8 X-0.5-21-55O 23 54 718

9 X-0.5-21-100O 23 101 771

10 X-0.5-21-160O 23 160 828

11 (control) X-0.5-21-550O 23 560 1222

12 (+cap plate) X-0.5-21-55C 23 55 731

Note: measured β = 0.50 for all specimens. 196

3.2. Test Setup and Instrumentation 197

The experiments were performed using a 5000kN-capacity Baldwin testing frame. The connection was 198

supported on a pedestal, which was secured to the laboratory floor. A steel collar was fitted around the branch at 199

the top to prevent any lateral movement and the steel collar was welded to a steel plate, which was then bolted to 200

the machine head. The branch could then be centred vertically with the load application and support points. The 201

X-connection was thus tested under direct transverse compression rather than as a simply supported T-202

connection, which removed any chord normal stress effects on the connection results and hence Qf = 1.0 203

(discussed further by Packer et al., 2017). This configuration also enabled the end distance to be varied for the 204

different test specimens. 205

All connections were instrumented with a linear variable differential transformer (LVDT) placed between 206

the top of the pedestal and the bottom of the machine head to monitor the total branch elastic deformation and 207

connection local deformation. A system, which consisted of a K610 optical camera that recorded the global 208

coordinates of light-emitting-diode (LED) targets positioned on the test specimens, was used to measure the 209

vertical connection displacements and chord face deformation profiles. For four connections, when the branch 210

was very close to the chord end (e = 25mm) and when the branch was in the middle of the chord length (control), 211

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for both HSS chord sizes, strain gauges were installed around the branch perimeter to measure the non-uniform 212

strain distribution close to the connection and the bending moment distribution along the branch height. Typical 213

test setups are shown in Fig. 10. 214

(a) Specimen with e = 25mm (b) Control specimen

Fig. 10. Typical test setups 215

Experimental Results 4.216

4.1. Geometric and Material Properties 217

Cross sections of the RHS branch and chord members were cut with a thickness of 1 in., machined, and 218

traced in AutoCAD to obtain the outside dimensions, inside/outside corner radii, as well as the cross-sectional 219

areas of the flat faces, corners, and full sections of the RHS members. The wall thicknesses were measured with 220

a 1-in. Mitutoyo micrometer. The average geometric properties are given in Tables 2 and 3. Since the measured 221

thickness of the chord members exceeds the 5% tolerance on the wall thickness for CSA G40 grade, it was likely 222

made to ASTM A500 grade. 223

Table 2. Average measured RHS cross-sectional dimensions 224

Designation Width and height Wall thickness Corner radius

[mm] h and b [mm] t [mm] Out., ro [mm] In., ri [mm]

HSS 203x203x6.35 Chord 203.6 5.96 14.98 8.99

HSS 203x203x9.53 Chord 203.1 8.85 23.82 14.98

HSS 102x102x9.53 Branch 101.7 8.74 18.62 9.68

225

226

227

228

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Table 3. Measured RHS cross-sectional areas 229

Designation [mm] 2γ Cross-sectional area, A [mm

2]

Flat Corner Total

HSS 203x203x6.35 Chord 34 4046 446 4492

HSS 203x203x9.53 Chord 23 5405 1090 6495

HSS 102x102x9.53 Branch 12 2225 755 2980

Standard longitudinal tensile coupons, taken from the three flat sides at least 90º from the weld seam and 230

from three corners of each of the two RHS chord members used for the connection tests, were cut, machined, 231

and tested in accordance with ASTM E8 (2013). The average material properties, including the yield strength 232

(fy), ultimate strength (fu), and rupture strain (εrup) are summarized in Table 4. 233

The strengths of the corners are increased relative to the flats (the yield strengths are increased significantly 234

and the ultimate strengths are increased moderately), while the ductility in the corners is reduced (the rupture 235

strains are reduced) due to the greater amount of cold-working. Typical stress-stain plots for the flats and corners 236

of the HSS 203x203x9.53 chord member are shown in Fig. 11 (a) and (b), respectively. 237

Table 4. Average measured RHS chord material properties 238

Designation [mm] fy [MPa]* fu [MPa] εrup

HSS 203x203x6.35 flat 388.8 508.7 0.287

HSS 203x203x6.35 corner 540.5 598.9 0.176

HSS 203x203x9.53 flat 398.1 519.8 0.306

HSS 203x203x9.53 corner 513.8 604.4 0.219

* Yield strength determined by the 0.2% offset method. 239

240

(a) HSS 203x203x9.53 flat coupons (b) HSS 203x203x9.53 corner coupons

Fig. 11. Typical stress-strain curves from tensile coupon tests 241

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4.2. Connection Test Results 242

Testing was quasi-static and displacement-controlled to connection failure. Failure of all 12 connections was 243

governed by the chord plastification limit state, which was exhibited by flexural deformation of the chord 244

connecting face. For the connection with a cap plate, the chord face eventually failed in punching shear at the 245

junction with the cap plate. Typical failure modes are illustrated in Fig. 12. 246

(a) e = 25mm (b) e = 55mm (+cap plate)

Fig. 12. Typical failure modes 247

To obtain the connection load-deformation curve, the branch load was measured by the testing machine’s 248

load cell and vertical connection deformation was determined from the global vertical displacement of the LED 249

located close to the connection (on the branch, 50mm above the chord face), which also represented the local 250

deformation because the bottom of the HSS chord was restrained from vertical displacement. Typical load-251

displacement behaviours are illustrated in Fig. 13 (a) and (b) for the HSS chords with 2γ = 34 and 23, 252

respectively. 253

The ultimate connection capacities of all 12 tests were controlled by the 3% b0 deformation limit. The 254

connections exhibited considerable deformation and ductility beyond this point, and strength continued to 255

increase with strain hardening of the HSS material and membrane action in the chord. After punching shear 256

occurred for the connection with a cap plate, the load dropped but the cap plate continued to provide restraint to 257

the chord sidewalls. 258

Based on the plots, it was found that when the connection is very close (24-25 mm) to an open chord end, 259

there is a reduction in strength of approximately 27-29% compared to the regular connection without chord end 260

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effects. Alternatively, if the chord end is welded with a cap plate, there is an increase in strength of about 27-261

30% compared to the regular connection. 262

(a) HSS chord with 2γ=34 (a) HSS chord with 2γ=23

Fig. 13. Typical load-displacement curves 263

The local vertical deformation along the centerline of the top chord face was processed from the LED 264

readings. Typical graphs of these profiles, for various branch loads, are presented in Fig. 14 for the 2γ = 34 HSS 265

chord. The theoretical transverse yield lines at a distance of 00.5 1b −β , modified slightly to account for the 266

weld size w, from the edges of the branch are also plotted. 267

It is shown that deformations were concentrated near the connection and extended for a length of 268

approximately 400mm (≈2b0) and 300mm (≈1.5b0), from the branch edge for the 2γ = 34 and 23 HSS chords, 269

respectively, beyond which there is almost no deformation. This indicates that the chord lengths used, based on 270

the EN 1993-1-8 formula, are conservative in excluding any chord end effects on the connection results. 271

(a) Specimen with e = 25mm (b) Control specimen

Fig. 14. Typical chord face deformation profiles for HSS chord with 2γ = 34 272

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The normal stress distributions around the branch perimeter were processed from the strain gauge readings. 273

Typical plots of these distributions just above the chord face, at the connection load of N1,3%, are presented in 274

Fig. 15 for the HSS chord with 2γ = 34. The branch remained elastic under all applied loads, with the stresses 275

being well below the nominal yield stress of 345MPa. 276

It is observed that the branch stress distribution is highly non-uniform near the connection. Maximum 277

compressive stress is seen in the two branch walls parallel to the chord, while minimum compressive stress (or 278

even tensile stress in this case) is seen in the two branch walls transverse to the chord. This shows that the axial 279

load is resisted primarily by the two branch walls parallel to the chord, due to the stiffness of the HSS chord 280

walls. The two branch walls transverse to the chord are much less effective in resisting the load, because of the 281

flexibility of the HSS chord face. 282

(a) Specimen with e = 25mm (N1,3% = -91kN) (b) Control specimen (N1,3% = -127kN)

Fig. 15. Typical branch stress distribution close to the connection at N1,3% for HSS chord with 2γ = 34 283

The tensile stresses seen in the branch transverse walls are likely due to interference afforded by the branch 284

against chord face deformation. Specifically, as the chord face, which is supported at the chord walls, attempts to 285

deform in the transverse direction, it is restrained by the branch. Therefore, the branch pulls up, through the 286

weld, on the chord face; and the chord face, through the weld, pulls down on the branch, resulting in the 287

observed tensile stresses in the middle of the transverse branch walls, as illustrated in Fig. 16. 288

The differences in the stresses between the transverse walls suggest the presence of branch in-plane bending, 289

which may have been caused by branch initial out-of-straightness (for both the offset and regular connections) as 290

well as rotation applied at the branch base and lateral force induced at the branch top, thus leading to higher 291

tensile stresses in the transverse wall closest to the open chord end (for the offset connections). Nevertheless, 292

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comparable compressive stress distribution along the branch longitudinal walls is seen for both the offset 293

connections (e = 25mm) and the regular connections (control). 294

295 Fig. 16. Geometric compatibility for RHS-to-RHS connection 296

Evaluation of Results 5.297

5.1. Comparison with Yield Line Predictions 298

In order to evaluate the proposed analytical models, the experimentally-determined yield loads N1,3% (the 299

loads at the 3% b0 ultimate deformation limit) are plotted against the end distance e and compared with the 300

different yield line predicted strengths, calculated using measured chord yield strengths (fy0), based on the chord 301

flat faces, and with effective dimensions for the chord width (b0/), branch width (b1

/), width ratio (β

/), and end 302

distance (e/). The average weld sizes were used in the calculations. The comparisons for the two sets of six tests 303

using the HSS 203x203x6.35 and HSS 203x203x9.53 chords, with an actual chord slenderness (2γ) of 34 and 23, 304

are presented in Fig. 17 (a) and (b), respectively. 305

The calculation using effective dimensions is more accurate and increases the yield load value. Since the 306

yield line method is an upper bound method, mechanism no. 1, which computes the lowest yield load, is the 307

critical mechanism. Mechanisms no. 2 and 3, plotted respectively in Fig. 17 for m = 3 and x1/ = b1

/ to match 308

experimental observations (Fig. 18 (c) and (d)), never govern. m = 3 is also used to plot emin/b0 vs. β in Fig. 7 309

using Eq. (8), but it is again not critical. The plots show that the experimental loads agree well with the resulting 310

predicted loads and demonstrate a similar trend. Specifically, for end distances less than approximately emin (Eq. 311

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(4), the yield line solution based on mechanism no. 1), the strengths of connections near an open chord end are 312

reduced relative to their “regular connection” counterparts. For end distances greater than about emin, the 313

strengths of such connections reach those of their regular counterparts (control specimens). While the test results 314

are very close to the predicted PY1/ values for the 2γ = 34 chord (Fig. 17 (a)), they are slightly higher (but within 315

10%) for the 2γ = 23 chord (Fig. 17 (b)), which indicates that mechanism no. 1 is still conservative. Thus, 316

overall, the comparison justifies yield line model no. 1 as an accurate predictor of the yield load for the offset 317

connections, with a mean actual/predicted strength of 1.04 and an associated coefficient of variation (COV) of 318

0.035. Results for connections with a cap plate are not included in the statistical calculation. It is also observed 319

that emin does not seem to be affected by the chord wall slenderness, based on the trends for the two HSS chords. 320

(a) HSS Chord with 2γ=34 (fy0 = 389MPa) (b) HSS Chord with 2γ = 23 (fy0 = 398MPa)

Fig. 17. N1,3% or PY vs. e (using effective dimensions) 321

As designers do not know the weld sizes until the connections have been detailed, design recommendations 322

typically use full dimensions which do not account for the chord wall thickness t0 and weld size w in the 323

calculations. The predicted yield line strengths are thus re-calculated using measured chord yield strengths (fy0), 324

with full dimensions for the chord width (b0), branch width (b1), width ratio (β), and end distance (e). The 325

resulting computed values for mechanism no. 1 (PY1) are found to be always less than the experimental results 326

(N1,3%) for both the 2γ = 34 and 23 chords, with an overall actual/predicted mean value of 1.29 and a 327

corresponding COV of 0.053. This shows that yield line model no. 1, used in conjunction with full (or nominal) 328

dimensions as a simplification, is a conservative predictor of the yield load for the offset connections, and can 329

thus be used for design. This also validates the corresponding yield-line solution for the minimum end distance, 330

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min 0 1e b= −β , to develop the full connection strength, provided β ≤ 0.85 and chord plastification is the 331

governing failure mode. 332

5.1.1. Observed Yield Line Patterns 333

Application of the analytical yield line patterns to the deformed chord face for four of the tested connections 334

is shown in Fig. 18. While the rectilinear yield lines can only approximate the curved deformed surface, the 335

general yield line mechanisms can still be seen. The traditional mechanism is associated with the control 336

specimen with a large e, and mechanisms no. 1, 2, and 3 are observed for the end connections, with e ≤ 55mm 337

(shown for e = 55mm), e = 100mm, and e = 160mm, respectively. 338

(a) Traditional mechanism (control specimen) (b) Mechanism 1 (e = 55mm)

(c) Mechanism 2 (e = 100mm) (d) Mechanism 3 (e = 160mm)

Fig. 18. Observed yield line patterns for HSS chord with 2γ = 34 339

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5.2. Evaluation of EN 1993-1-8 Amendments 340

In the proposed amendment to EN 1993-1-8, an end distance greater than the maximum of (2γ/10)d0 or 341

2.5d0, measured from the edge of the branch, is required for CHS connections, in order for the strengths to be 342

predicted by traditional design equations. To assess the applicability of this end distance rule for RHS 343

connections, the diameter (d0) of the CHS chord is replaced by the width (b0) of the square HSS chord studied in 344

this research. Substituting 2γ = 34 and 23 for the two HSS chords gives a minimum required end distance of 345

3.4b0 and 2.5b0, respectively. This is much larger than that predicted by the yield-line solution of 0 1b −β or 346

0.71b0, computed with β = 0.5 for both chords, which was found to agree well with experimental results. Thus, 347

the EN 1993-1-8 formula is excessively conservative, and not particularly suitable for estimating the minimum 348

required connection end distances for open-ended chord members used in RHS trusses and girders. 349

The end connections found in practical applications are typically asymmetrical about the branch. However, 350

the research on end distance effects, on which the EN 1993-1-8 formula is based, examined connections that are 351

symmetrical about the branch. Thus, the recommended EN end distance or chord length is specifically relevant 352

for the design of isolated connections in experimental or numerical tests, in order to develop the traditional yield 353

line mechanism and exclude the influence of chord boundary conditions on the connection strength. In this 354

respect, based on the chord face deformation profiles on both sides of the branch for the regular connections, or 355

on the long-chord side of the branch for the offset connections, a chord length of approximately 2.0b0 and 1.5b0 356

was determined to be adequate for the two HSS chords with 2γ = 34 and 23, respectively (see Fig. 14 for the 357

2γ = 34 chord profiles). These observed values are still less than 3.4b0 and 2.5b0, calculated above, as the 358

maximum of (2γ/10)b0 or 2.5b0. Hence, the EN 1993-1-8 formula is reasonably conservative, and still suitable 359

for determining the minimum required chord lengths of RHS chord members used for isolated “symmetrical” 360

connections in experimental or numerical research. 361

5.3. Influence of Providing a Chord End Cap Plate 362

As illustrated by the load-displacement curves in Fig. 13 for both HSS chords, adding a cap plate to the 363

chord end restrains chord deformation and increases the connection stiffness. As such, the connection loads at 364

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the 3% b0 deformation limit exceed that of the control specimens, thus indicating that welding a cap plate to the 365

chord end is effective in achieving the full strength of a regular connection. 366

5.4. Weld Effective Length for Offset Connections 367

Welds may be designed to resist the actual forces in the branch. This requires the use of weld effective 368

lengths, which account for the non-uniform load transfer around the weld perimeter. Suitable effective lengths 369

for RHS T-, Y-, or X-connections subjected to a branch axial load are provided in Table K5.1 of AISC 360-16 370

(Eq. K5-5) and illustrated in Fig. 19 (a). The effective length is given by: 371

(13) 1

1

22

sine e

hl b= +

θ (Regular RHS connection) 372

Both Eq. (13) and Fig. 19 (a) show that, for a regular RHS connection associated with a large end distance, it 373

is primarily the two walls or the two welds in the longitudinal direction which are considered effective in 374

resisting the applied loads. This agrees with the branch stress distribution observed for the control specimens at 375

the connection load of N1,3% (see Fig. 15 (b) for the 2γ = 34 chord). In Eq. (13), an effective weld length in the 376

transverse direction, taken the same as the effective branch width be (given by Eq. K1-1 of AISC 360-16) used 377

for checking local yielding of a branch, is also included, where be/2 cannot exceed b1/4 when β > 0.85 or 378

θ1 > 50º. Thus, it is acknowledged that a portion of the branch or weld transverse lengths may be considered 379

effective. 380

(a) Regular connection

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(b) Offset connection

Fig. 19. Effective weld lengths for RHS-to-RHS X-connections (θ1 = 90o) 381

As described earlier, the offset connections with e = 25mm show a similar compressive stress distribution 382

along the branch longitudinal walls, at the connection yield load, as the control specimens for both HSS chords 383

(compare Fig. 15 (a) with (b) for the 2γ = 34 chord). For the offset connections, the tensile stresses seen in the 384

branch transverse wall closest to the open chord end may render the transverse weld length be along this side to 385

be ineffective. Thus, for the case of branch axial loading, the weld effective length developed for a regular RHS 386

connection may be applied, with a slight modification, to an offset RHS connection, as presented by Eq. (14) and 387

Fig. 19 (b). 388

(14) 1

1

2

sine e

hl b= +

θ (Offset RHS connection) 389

Conclusions and Recommendations 6.390

Based on the results for the 12 connection tests, shown in Fig. 20, the following conclusions and 391

recommendations are made for RHS-to-RHS axially-loaded T-, Y-, and X-connections: 392

• When a connection has a small end distance, e, to an open chord end, a modified yield line mechanism 393

controls connection failure and produces a connection strength lower than that associated with “regular” 394

connections (remote from the chord ends). The proposed closed-form analytical model herein, which is 395

evaluated against the experimentally-determined 3% b0 deformation limit load, is shown to be an 396

accurate predictor of the yield load PY for “offset” connections (near an open chord end), and PY can be 397

conservatively estimated from: 398

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( )

20 0 1

1

0 0 1

22sin 2 1

1 sin

y

Y f

f t heP Q

b b

θ = + + −β −β θ

for e < emin 399

where min 0 1e b= −β represents the critical end distance at which full connection capacity is achieved. 400

This minimum end distance, emin, has also been adopted in Table K3.2A of AISC 360-16, as a 401

requirement for the application of traditional design rules for RHS connections. Alternatively, an end 402

distance equal to the chord member width, or emin = b0, is conservative. 403

• The proposed amendment to EN 1993-1-8, requiring an end distance as the maximum of (2γ/10)b0 or 404

2.5b0 in order to achieve full connection strength, is deemed excessively conservative and not suitable 405

for open-ended RHS chord members in real structures. It is, however, considered suitable (and 406

conservative) for determining the minimum chord lengths of RHS chord members used for isolated, 407

“symmetrical” connections in experimental or numerical tests. 408

• When a connection is closer to an open chord end than emin, reducing the strength predicted by 409

traditional yield line equations for RHS connections by 50%, in lieu of a detailed calculation, as 410

recommended by the AISC 360-16 Commentary, is safe for design. 411

• Stiffening the chord end with a cap plate is effective in developing the strength of the connected face 412

against chord plastification. A minimum end distance, emin, does not apply for this case. 413

• The branch member in an end connection needs to be laterally restrained (which is typically the case 414

when it is connected to the rest of the frame in a real structure) to prevent lateral movement at the branch 415

top, due to a rotation applied at the branch base. 416

• The weld effective lengths provided for a regular connection, which are used to proportion welds for the 417

actual forces in the branch, can also be applied to an offset connection with a slight modification of 418

1 12 sine el h b= θ + , for the case of branch axial loading. 419

The conclusions summarized above, which are all associated with the chord plastification limit state, only 420

apply if β ≤ 0.85. They can be extended to T- and Y-connections, since flexure of the chord connecting face 421

should produce the same effects in these connections as in the X-connections studied in this research. For a Y-422

connection with θ1 < 90º, the connection capacity depends on the component of the branch load normal to the 423

chord, PYsinθ1, with a branch footprint of h1/sinθ1, which has been accounted for in the equations above. 424

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6.1.1. Application to Other Connection Types 425

The minimum end distance developed for the RHS-to-RHS X-connections should be applicable to the 426

following connection types, for the case of branch axial loading: 427

• Gapped and overlapped K- and N-connections, which consist of a compression branch member pushing 428

into the chord face and a tension branch member pulling out. Compared to X-connections, the resulting 429

chord face deformations are expected to be less. 430

• Longitudinal plate-to-RHS connections, as the same limit state of chord plastification governs. For these 431

plate-to-RHS connections, β would equal t1/b0 (typically varying between 0.05 to 0.25) and the end 432

distance would be min 0 1 01e b t b= − , where t1 is the plate thickness. 433

• Transverse plate-to-RHS connections, which are generally not governed by chord plastification for 434

practical values of β and low compressive stresses in the chord connecting face. However, if either local 435

yielding of the plate or punching shear in the chord is critical, the associated local deformations are 436

expected to be more confined than those produced by chord face plastification. 437

438 Fig. 20. 12 tested specimens 439

Acknowledgements 440

The authors are grateful for the financial support provided by the Canadian Institute of Steel Construction 441

(CISC) and the Natural Sciences and Engineering Research Council of Canada (NSERC), as well as the in-kind 442

donation of HSS materials by Atlas Tube Inc. and the laboratory assistance of Mr. F. Wei. 443

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References 444

AISC. 2010. Specification for structural steel buildings. ANSI/AISC 360-10 Chapter K, American Institute of 445

Steel Construction, Chicago, USA. 446

AISC. 2016. Specification for structural steel buildings. ANSI/AISC 360-16 Chapter K, American Institute of 447

Steel Construction, Chicago, USA. 448

ASTM. 2013. Standard test methods for tension testing of metallic materials. ASTM E8/E8M-13a, American 449

Society for Testing and Materials, West Conshohocken, USA. 450

ASTM. 2014. Standard specification for cold-formed welded and seamless high-strength, low-alloy structural 451

tubing with improved atmospheric corrosion resistance. ASTM A847/A847M-14, American Society for 452

Testing and Materials, West Conshohocken, USA. 453

CEN. 2005. Eurocode 3: Design of steel structures – Part 1-8: Design of joints. EN 1993-1-8:2005(E), European 454

Committee for Standardization, Brussels, Belgium. 455

CSA. 2013. General requirements for rolled or welded structural quality steel. CSA G40.20-13/G40.21-13, 456

Canadian Standards Association, Toronto, Canada. 457

IIW. 2012. Static design procedure for welded hollow section joints – Recommendations, 3rd ed. IIW Doc. XV-458

1402-12, International Institute of Welding, Commission XV, Paris, France. 459

ISO. 2013. Static design procedure for welded hollow-section joints – Recommendations. ISO 14346:2013(E), 460

International Organization for Standardization, Geneva, Switzerland. 461

Kosteski, N., and Packer, J.A. 2003a. Welded tee-to-HSS connections. Journal of Structural Engineering, 462

American Society of Civil Engineers, 129(2): 151-159. doi:10.1061/(ASCE)0733-463

9445(2003)129:2(151). 464

Kosteski, N., and Packer, J.A. 2003b. Longitudinal plate and through plate-to-hollow structural section welded 465

connections. Journal of Structural Engineering, American Society of Civil Engineers, 129(4): 478-486. 466

doi:10.1061/(ASCE)0733-9445(2003)129:4(478). 467

Kosteski, N., Packer, J.A., and Puthli, R.S. 2003. A finite element method based yield load determination 468

procedure for hollow section connections. Journal of Constructional Steel Research, 59(4): 453-471. 469

doi:10.1016/S0143-974X(02)00066-4. 470

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Lu, L.H., de Winkel, G.D., Yu, Y., and Wardenier, J. 1994. Deformation limit for the ultimate strength of hollow 471

section joints. In Proceedings of the 6th International Symposium on Tubular Structures, Melbourne, 472

Australia, pp. 341-347. 473

Packer, J.A., and Henderson, J.E. 1997. Hollow structural section connections and trusses – a design guide, 2nd 474

ed. Canadian Institute of Steel Construction, Toronto, Canada. 475

Packer, J.A., Wardenier, J., Zhao, X.L., van der Vegte, G.J., and Kurobane, Y. 2009. Design guide for 476

rectangular hollow section (RHS) joints under predominantly static loading. CIDECT Design Guide No. 477

3, 2nd ed. CIDECT, Geneva, Switzerland. 478

Packer, J.A., Puthli, R., van der Vegte, G.J., and Wardenier, J. 2017. Discussion on the paper, “Experimental and 479

numerical assessment of RHS T-joints subjected to brace and chord axial forces”, by Nizer et al. Steel 480

Construction, 10(1): 89-90. doi:10.1002/stco.201770109. 481

van der Vegte, G.J., and Makino, Y. 2006. Ultimate strength formulation for axially loaded CHS uniplanar T-482

joints. Int. Journal of Offshore and Polar Engineering, ISOPE, 16(4): 305-312. 483

van der Vegte, G.J., and Makino, Y. 2010. Further research on chord length and boundary conditions of CHS T- 484

and X-joints. Advanced Steel Construction, 6(2): 879-890. 485

Voth, A.P., and Packer, J.A. 2012a. Branch plate-to-circular hollow section connections. II: X-type parametric 486

numerical study and design. Journal of Structural Engineering, American Society of Civil Engineers, 487

138(8): 1007-1018. doi:10.1061/(ASCE)ST.1943-541X.0000545. 488

Voth, A.P., and Packer, J.A. 2012b. Numerical study and design of T-type branch plate-to-circular hollow 489

section connections. Engineering Structures, 41: 477-489. doi:10.1016/j.engstruct.2012.03.034. 490

Wardenier, J., Packer, J.A., Zhao, X.L., and van der Vegte, G.J. 2010. Hollow sections in structural applications, 491

2nd ed. CIDECT, Geneva, Switzerland. 492

Zhao, X.L. 2000. Deformation limit and ultimate strength of welded T-joints in cold-formed RHS sections. 493

Journal of Constructional Steel Research, 53(2): 149-165. doi:10.1016/S0143-974X(99)00063-2. 494

495

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List of Symbols 496

A Cross-sectional area (mm2)

be Effective branch width (mm)

b0, b0/ Nominal/full (out-of-plane) width of RHS chord, effective width (= b0 – t0) (mm)

b1, b1/ Nominal/full (out-of-plane) width of RHS branch, effective width (= b1 + 2w) (mm)

d0 Outside diameter of CHS chord (mm)

E Modulus of elasticity (MPa)

e, e/, emin End distance from edge of branch to chord end, measured from edge of weld (= e – w), minimum

required end distance (mm)

fy, fy0 Yield stress of material, yield stress of RHS chord (MPa)

fu Ultimate stress of material (MPa)

h0 Outside (in-plane) depth of RHS chord (mm)

h1, h1/ Nominal/full (in-plane) depth of RHS branch, effective depth (= h1/sinθ1 + 2w) (mm)

le Effective weld length (mm)

l0 Chord length (mm)

m Mechanism 2: slope of inclined yield lines from branch side to open chord end

N1 Branch axial load (N)

N1,3% Connection load at an ultimate deformation of 3% b0 (N)

PY, PY/ Connection load predicted by yield line mechanisms using full, effective dimensions (N)

Qf Chord stress function

t0, t1 Wall thickness of HSS chord, wall thickness of HSS/plate branch (mm)

w Measured fillet weld size (horizontal leg length) (mm)

x, x/ Mechanism 3: distance required for inclined yield lines from edge of branch to open chord end to

converge, measured from edge of weld (= x – w) (mm)

β, β/ Nominal/full width ratio (= b1/b0 for RHS), effective width ratio (= b1

’/b0

’ for RHS)

δ Vertical connection displacement (mm)

εrup Rupture strain of material

2γ Chord width (diameter)-to-thickness ratio (= 2b0/t0 for RHS, = 2d0/t0 for CHS)

θ1 Included angle between branch and chord (º)

497

498

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List of Figure Captions 499

Fig. 1. RHS-to-RHS connection near a truss open end, with chord stiffened 500

Fig. 2. Proposed end distances in EN 1993-1-8 (shown for CHS) 501

Fig. 3. Yield line mechanism for a regular RHS-to-RHS axially-loaded X-Connection 502

Fig. 4. Yield line mechanism No. 1 for an offset RHS-to-RHS axially-loaded X-connection 503



Fig. 7. emin/b0 vs. β based on postulated yield line mechanisms 506

Fig. 8. Failure criteria (adapted from Wardenier et al., 2010) 507

Fig. 9. Square HSS-to-HSS X-connection test specimens 508

Fig. 10. Typical test setups 509

Fig. 11. Typical stress-strain curves from tensile coupon tests 510

Fig. 12. Typical failure modes 511

Fig. 13. Typical load-displacement curves 512

Fig. 14. Typical chord face deformation profiles for HSS chord with 2γ = 34 513

Fig. 15. Typical branch stress distribution close to the connection at N1,3% for HSS chord with 2γ = 34 514

Fig. 16. Geometric compatibility for RHS-to-RHS connection 515

Fig. 17. N1,3% or PY vs. e (using effective dimensions) 516

Fig. 18. Observed yield line patterns for HSS chord with 2γ = 34 517

Fig. 19. Effective weld lengths for RHS-to-RHS X-connections (θ1 = 90⁰) 518

Fig. 20. 12 tested specimens 519

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